| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 This is a standard FP2 loci question requiring algebraic manipulation to identify a circle from |z-3|=2|z|, recognizing that |z+3|=|z-i√3| represents a perpendicular bisector, sketching both loci, and shading a region. While it involves multiple parts and some algebraic work, these are routine techniques for Further Maths students with no novel insight required—slightly easier than average A-level difficulty when considering the full spectrum. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
The point $P$ represents a complex number $z$ on an Argand diagram such that
$$|z - 3| = 2|z|.$$
\begin{enumerate}[label=(\alph*)]
\item Show that, as $z$ varies, the locus of $P$ is a circle, and give the coordinates of the centre and the radius of the circle.(5)
\end{enumerate}
The point $Q$ represents a complex number $z$ on an Argand diagram such that
$$|z + 3| = |z - i\sqrt{3}|.$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Sketch, on the same Argand diagram, the locus of $P$ and the locus of $Q$ as $z$ varies.(5)
\item On your diagram shade the region which satisfies
$$|z - 3| \geq 2|z| \text{ and } |z + 3| \geq |z - i\sqrt{3}|.$$ (2)
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2008 Q10}}